Factors causing the MCF
The popular description of the causes of the MCF is that it results from a misconception or non-understanding of the notions of randomness and independence. Nothing is more true than that; however, understanding these two concepts only by their lexicographic definition, and even the probabilistic/statistic definition of independence of events, does not suffice in either explaining the formation of the fallacy or correcting it. Explaining a cause scientifically should automatically provide the tools for eliminating it and correcting the fallacy, and the core element for both explanation and correction is the concept of understanding specific to the context.
Obviously, the outcomes that trigger the MCF are random events. In probabilistic/statistical terms, they are random because they are the result of trials of the same general random experiment: Spinning the roulette wheel (the random experiment) is done several times (the trials), which means that they are independent of each other. Some may think that they are not independent, since they are the results of the same experimental setup (entering the causes of MCL), but independence of events must be understood in a statistical sense here and not in that of physical causality: The trials as actions may not be independent as performed by the same person or machine, but their outcomes, as elementary events, are independent because the possibility (or probability, if you like) of occurrence of one does not depend on another.
This statistical independence comes from the premise that the outcomes as elementary events are equally possible because all the physical factors of the experiment have been objectively ignored. In other words, the experiment and its associated events are random. Of course, independence relates to randomness and probability theory does not define randomness. Once we have understood the notion of statistical independence, we have to clear up the concept of randomness in this MCL context.
The lack of this knowledge about independence and randomness falls within the educational factors that cause MCL. However, the issue is not only that of acquiring knowledge, but also of perception, as we shall see further. For the moment, let’s stay with randomness.
We have seen that randomness is both order and disorder, and for adequately understanding it, we must not ignore either of these two attributes. When a player affected by MCF expects a succession of non-favorable outcomes to end as having a reason for that, that person actually is strongly inclined to take randomness as order, because the reason behind that expectation is the belief that what seems to be disorder must be restored as order – or at least come to the tendency of restoring. The order in this case is expressed through either the mathematical probability of that event or any registered, memorized, or known average frequency of that event in the past experience of the player or of other players.
In the case that the player knows the probability of the event, his or her expectation is for the relative frequency of that event over his or her plays or the observed activity of the game in question to approach this probability. In the opposite case, the expectation is for the current relative frequency to match an average relative frequency recorded statistically in player’s own past games or the history of that game.
Equating probability with relative frequency over the short-to-medium run is obviously a mathematical error. The mathematical result that links probability and relative frequency is the Law of Large Numbers (LLN), in the sense that probability is the limit of the sequence of the relative frequencies calculated when the number of trials increases. Equating the current relative frequency with an average based on a large enough number of past recordings is a mathematical error as well, as LLN does not provide such kind of relationship – it is only about the limit, and the terms of a sequence may differ from each other in whatever range over a finite interval without affecting the limit of the sequence.
LLN grants randomness a “minimal” order in the sense of convergence, and our hypothetical player inclines to grant it even “more ordered” order, by translating the LLN result over a finite interval of trials or his or her short experience.
Summing up, the educational causes of the MCF consist of three interdependent fallacies, misconceptions, and errors:
1) employing physical causal (in)dependence instead of statistical (in)dependence in reasoning (fallacy);
2) taking randomness to be order and not disorder (misconception);
3) equating probability with relative frequency, (mathematical error).